3900_2011_PS4 - Problem Set 4 Chem 3900: Physical Chemistry...

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Unformatted text preview: Problem Set 4 Chem 3900: Physical Chemistry II Spring 2011 Due: Class, Friday, Feb 25th Please make sure to write your name and the recitation you’ll attend. When you print out your work, try to minimize the number of pages. You can do this by printing multiple pages per sheet and/or by using a duplex printer. Problem 1. A sample of sucrose, C12H22O11, was mass 0.5034 g is burned in a bomb initially at 298K. The temperature is observed to increase by 1.357K. It was also observed that adding 9549 J of electrical heat increased the temperature by 1.689K. i) Determine ∆ ii) Use the data in Table 19.2 to calculate ∆ the results. 298 and ∆ 298 for the combustion of sucrose. 298 for the combustion of sucrose. Compare Problem 2. Make a big table of the following quantities: , , ∆ , ∆ , ∆ for the following classical reversible processes for an ideal gas: isochoric (constant V) process , isobaric expansion (constant P), isothermal expansion (constant T), and adiabatic expansion (q = 0). In future homework sets we are going to add ∆ , ∆ , ∆ . You can assume that is constant. For the quantities you have already calculated you don’t need to include the derivation. Problem 3. A gas obeys the viral equation of state obtained by expanding the Redlick‐Kwong compressibility factor in powers of 1/ V and keeping only terms up to and including 1/ V (i.e. keeping only the second virial coefficient in the expansion). i) First calculate the second virial coefficient of the Redlick‐Kwong equation of state. ii) Suppose that n moles of this gas undergo an isothermal, reversible change in volume from volume V1 to volume V2. Calculate the work for this process iii) Consider your answer to (ii) for the case of a compression. At low temperatures (as low as you please), compare the work to that associated with compressing an ideal gas: is it greater, less, or the same? Interpret your answer in terms of intermolecular forces. Repeat this 1 analysis for high temperatures (as high as you want), together with a molecular interpretation. Consider the particular case of ethane. Take V 1 = 60 L mol‐1, V 2 = 20 L mol‐1. Plot the difference between the work of compression for the RK gas and an ideal gas over the temperature range 400 § T § 1600 K. Is it consistent with your result in (iii)? iv) Problem 4. In addition to his celebrated theories of special and general relativity, Albert Einstein made many other contributions to chemistry and physics. One of these is his theory of the heat capacity of crystalline solids, described in Example 17‐3 and Exercise 17‐20 of McQ&S. In the Einstein model of diamond, the vibrational motion of each carbon atom along x, y, and z directions is represented as a harmonic oscillator. The energy level of the harmonic oscillator was discussed in Chem 3890 (It is also shown in Figure 5.7 of McQ&S). In a collection of harmonic oscillators at temperature T, different oscillators occupy different quantum states, labeled by the quantum number v. The average value of this quantum number, <v >, is given by v vPv 0 with Pv the Boltzmann probability that the vibration has quantum number v. i) Calculate < v > The vibrational frequency of diamond is = 2.75 ä 1013 s‐1. Calculate the corresponding vibrational temperature. ii) iii) Use your answer to (i) to determine the temperature at which each vibration in diamond is, on average, in its first excited state. How does this compare to (ii)? iv) Plot < v > as a function of T from 0 K to 2000K. Compare your result with Figure 17.4 of McQ&S. Can you use your plot to explain the temperature dependence of molar heat capacity of diamond? 2 ...
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